Complex CKM from Spontaneous CP Violation Without Flavor Changing Neutral Current

We analyse the general constraints on unified gauge models with spontaneous CP breaking that satisfy the conditions that (i) CP violation in the quark sector is described by a realistic complex CKM matrix, and (ii) there is no significant flavor changing neutral current effects in the quark sector. We show that the crucial requirement in order to conform to the above conditions is that spontaneous CP breaking occurs at a very high scale by complex vevs of standard model singlet Higgs fields. Two classes of models are found, one consisting of pure Higgs extensions and the other one involving fermionic extensions of the standard model. We give examples of each class and discuss their possible embeddings into higher unified theories. One of the models has the interesting property that spontaneous CP violation is triggered by spontaneous P violation, thereby linking the scale of CP violation to the seesaw scale for neutrino masses.


I. INTRODUCTION
It is now becoming increasingly clear that the dominant contribution to low energy CP violation arises from the complex CKM matrix which parameterizes the weak quark current coupling to the W-boson. Indeed the recent measurement [1] of the angle γ = −Arg(V ud V cb V * cd V * ub ) provides evidence [2] for a complex CKM matrix even if one allows for New Physics (NP) contributions to B d −B d mixing and B s −B s mixings.
However, this cannot be the full story of CP violation in elementary particle interaction [3] since it is believed that the explanation of the only cosmic manifestation of CP nonconservation i.e. the asymmetry between matter and anti-matter must come from sources other than the CKM CP violation; similarly the solution to the QCD θ problem may also imply new forms of CP violating interactions. Moreover, there is the fundamental question of the origin and nature of CP violation and its relation to other constituents and forces.
Even before the full story of CP violation is clear, one can ask the question as to whether the observed CKM CP violation is spontaneous in origin [4] or intrinsic to the Yukawa couplings in the theory. This question has nontrivial cosmological implications since spontaneous CP violation will lead to domain walls and in order to avoid conflict with observations such as WMAP data, one must have the scale of this breaking to be above that of the inflation reheating, thus imposing constraints on both cosmological as well as particle physics aspects of models.
In practical construction of models with spontaneous CP breaking, one must have one or more Higgs fields to have complex vevs [4]. It is obvious that implementing this requires extending the standard model, by having either more Higgs/or fermion fields plus Higgs because gauge invariance allows no room for Higgs vevs to be complex in the standard model. Furthermore, since spontaneous CP violation (SCPV) requires nontrivial constraints on the realistic gauge models, it is not surprising that the process of implementing it can lead to unpleasant side effects. One such unpleasant effect is the plethora of flavor changing neutral current (FCNC) effects induced in the process of obtaining spontaneous CP breaking. Therefore, the challenge in constructing realistic models with spontaneous CP violation is twofold: i) One should achieve genuine spontaneous CP violation and assure that the vacuum phase does lead to a non-trivially complex CKM matrix. This is not an easy task since CP invariance of the Lagrangian requires the Yukawa couplings to be real.
ii) One should find a natural suppression mechanism for FCNC in the Higgs sector.
Again, this is a challenging task, since there is in general a close connection [5] between the appearance of FCNC and the possibility of generating a complex CKM matrix through CP violating vacuum phases.
The above link between SCPV and FCNC can be seen by considering a two Higgs extension (φ 1,2 ) of the standard model to implement SCPV. It is well known (and we repeat the derivation in sec.II and in Appendix A) that general two Higgs models have FCNC mediated by neutral Higgs fields. In order to suppress these FCNC effects one may consider two possibilities. One consists of the introduction of extra symmetries which eliminate FCNC and guarantee natural flavour conservation ( NFC ) [6] in the Higgs sector. It is well known that the introduction of such symmetries in the two Higgs doublet framework eliminates the possibility of having spontaneous CP violation [5] . With three Higgs doublets one can have NFC and yet achieve spontaneous CP violation but the resulting CKM matrix is real, in contradiction with recent data. Above we have considered the case where FCNC are avoided through the introduction of extra symmetries, not by fine-tuning. It has been shown that even if one considers elimination of FCNC through fine-tuning, for three generations one cannot generate a realistic complex CKM matrix [7]. The other possibility for suppressing FCNC effects is by choosing a large mass for the neutral Higgs which violate flavour.
Indeed the strength of FCNC effects is proportional to 1/M 2 H where H denotes the new neutral Higgs field (we will denote the standard model Higgs by h). So clearly, suppression of FCNC effects require that M H become very large . On the other hand, as we show below, the magnitude of the CP phase (denoted by δ in the text) in this model is given by δ ∼ M W M H so that as M H → very large, δ → 0 and the theory becomes almost CP conserving. Note that to obtain CKM CP violation, we need δ ∼ 1. We will thus show that in the context of models with SCPV at the electroweak scale, it is not possible to obtain a complex CKM matrix while suppressing FCNC effects. In this class of SCPV models, obtaining a large CP phase and having significant FCNC seem to go together.
In this paper, we discuss the conditions under which this connection can be avoided. We point out that the crucial point is to have CP broken at a high energy scale. We present two classes of models: one where the extension involves only the Higgs sector of the standard model and another one which involves the fermion sector as well. In the latter case, there is a small departure from unitarity of the CKM matrix.
Several of the models we discuss have already been considered in the literature. We present a systematic classification of these models, adding some new ones and sharpening the connection between SCPV and FCNC. In particular, we present criteria for constructing realistic SCPV models free of FCNC constraints. This paper is organized as follows: in sec. II, we discuss the connection between SCPV and FCNC in doublet Higgs extension of the SM. In sec. III, we discuss spontaneous CP breaking at high scale in a pure Higgs extension and show how one can avoid the FCNC effects in this case. In sec. IV, we present a fermionic extension of the SM with spontaneous CP breaking at high scale. In sec. V, we discuss these two classes models into a left-right model and discuss two models one of which has the interesting property that spontaneous CP violation is triggered by spontaneous P violation. In sec. VI, we briefly comment on how our ideas can be extended to supersymmetric models and finally in sec. VII, we present our conclusions. In the appendices A and B we present a detailed demonstration of the results of sec. in sec. II and III.

II. TWO HIGGS DOUBLET MODEL FOR SCPV AND FCNC
The simplest extension of the standard model that can accommodate spontaneous CP violation is the two Higgs doublet model. If we denote the two Higgs doublets as φ 1,2 , and define V 0 (x, y) = −µ 2 1 x − µ 2 2 y + λ 1 x 2 + λ 2 y 2 + λ 3 xy, we can write the potential as follows: where We can now write down the potential in terms of the electrically neutral components of the doublets. It looks exactly the same as the above potential as long as we understand the various fields as the neutral components of the fields.
In order to discuss spontaneous CP violation [8], we look for a minimum of the form: The potential at this minimum looks like The three extremum equations are: Now let us study the implications of the extremum equations for SCPV and FCNC. Writing , it is straightforward to see that in general there will be FCNC mediated by neutral Higgs. We will consider next two possibilities for suppressing these FCNC. One involves the introduction of extra symmetries in order to implement Natural Flavour Conservation (NFC) [6] in the Higgs sector; the other considers the possibility of making very heavy the neutral Higgs which mediate FCNC. We will see that both possibilities do not work as far as generating a viable complex CKM matrix, but the discussion is useful in order to motivate the breaking of CP at a high energy scale which will be considered in sections 3 and 4.

A. Eliminating FCNC through extra symmetries
It is well known that it is possible to avoid FCNC by introducing for example a Z 2 symmetry which restricts the Yukawa couplings so that only one Higgs doublet gives mass terms to the down quarks while the other doublet gives mass to the up quarks. However, it has been shown [5] that the same symmetry which leads to these selective Yukawa couplings prevents the occurrence spontaneous CP breaking. A possible way out of this difficulty involves the introduction of a third Higgs doublet. In this case it is possible to obtain a CP violating vacuum [5] but the CKM matrix is real, in conflict with the recent experimental findings. The reason why CKM matrix is real in this case has to do with the fact that due to the selective Yukawa couplings, the vacuum phase which appears in the quark mass matrices can be eliminated by rephasing right handed quark fields.

B. Suppressing FCNC effects through large Higgs masses
It is straightforward to see that we could diagonalize one set of Yukawa couplings h u,d,1 so that the neutral Higgs (h) coming from the doublet φ 1 has flavor conserving couplings whereas that from φ 2 (H) has flavor violating couplings. In general of course the two neutral Higgs fields mix and therefore the h u,2 coupling which in the symmetry limit involves only the H Higgs field will have an admixture of the light Higgs h but mixing is is always proportional to the mass ratio m 2 h /M 2 H assuming M H ≫ m h . Thus FCNC processes will arise via the tree level exchange of H boson and will be proportional to M −2 H and a contribution from the mixing term which due to the mixing will also have the same kind of power dependence on M H . Therefore in order to suppress FCNC interactions, we must demand that M H be very large. This can be achieved by making −µ 2 2 > 0 and |µ 2 2 | ≫ v wk . Let us now study Eq. (6): this equation tells us the scale of the vev v 2 which depends on the scale of the mixing term µ 12 . (Note that getting the correct weak scale fixes µ 2 1 to be of order v wk and stopping FCNC tells us that |µ 2 2 | ≫ v 2 wk but so far µ 2 12 remains a free parameter.) We have two cases: (i) µ 2 12 ∼ v 2 wk and (ii) In case (i), it is easy to see using the middle equation above that: i.e. the vev of < φ 2 is highly suppressed in the limit of no FCNC. Note that the mass of the second neutral Higgs is not of order v 2 since in this case the vev is induced by a tadpole like diagram. Substituting this small value of v 2 in Eq.(c), we then see that for natural values of the parameters (λ i ), the only solution for the CP violating phase is δ = 0, π, ... .
On the other hand in case (ii), v 2 ∼ v wk but equation (7) above tells us that in this case also the expression in the bracket cannot give a nonzero δ since µ 2 12 v 1 v 2 ≫ 2λ 4 v 2 1 v 2 2 and the term within the bracket cannot vanish meaning that sinδ = 0 and hence no SCPV.
We therefore conclude that in this simple model, the requirement of suppression of the neutral current effects implies no SCPV. The main point is that to get a large enough SCPV phase, Eq(7) tells us that v 2 must be comparable in magnitude to v 1 . For this to happen, we must have |µ 2 2 | ∼ v 2 wk which again means that there must be large FCNC effects at low energies.
The above result can also be seen as follows: In a two Higgs doublet theory, one can change the basis of Higgs bosons to pass to a basis where the new doublets are and Φ 2 is the orthogonal combination to Φ 1 , where we have anticipated the vevs of the fields in the original basis, as discussed above. Now we see that < Φ 1 >= 0 while < Φ 2 > = 0 and it leads to the same mass matrices for quarks as before. Now we can choose parameters of the Higgs potential such that the mass of Φ 1 is very large to avoid FCNC effects. In this case, the effective theory below the mass of Φ 1 i.e. M Φ 1 is same as the standard model up to zeroth order in M W /M Φ 1 . Therefore, to this order, the vev of Φ 2 (which is the equivalent of the standard model Higgs) will be real, and there will be no spontaneous CP violation in the theory (to order M W /M Φ 1 ). This again proves that in the limit of zero FCNC, there will be no SCPV. In appendix A, we give explicit calculations in the mass basis that substantiates this conclusion.
This result can be generalized to the case of arbitrary number of Higgs doublets. For example for the case of three doublets, the argument is that as long as all the doublets couple to quark fields, at least two of the neutral Higgs bosons i.e. H 1,2 must be heavy in order to avoid large FCNC effects and this implies that |µ 2 23, | ≫ v 2 wk ; in that case their vev's must be suppressed and of order v 3 wk |µ 2 2,3 | and therefore small. The potential will then be forced to choose the minimum such that all SCPV phases are zero.

III. HIGH SCALE SPONTANEOUS CP VIOLATION LEADING TO COMPLEX CKM WHILE AVOIDING FCNC: MODEL WITH EXTRA HIGGS ONLY
In this section, we show how the FCNC problem is avoided if spontaneous violation of CP symmetry arises at a high scale. First we discuss this using a model with two SU(2) L ×U(1) Y Higgs doublets φ 1,2 as before and a complex singlet σ. The potential for this case can be written as follows: where V (φ 1,2 ) is defined in Eqs.(1), ( 3) and the other two terms are given by and It is clear that the minimum of the potential V (σ) corresponds to < σ >= Λe iα , where Λ ∼ M 0,1,2 ≫ v wk and α can be large. Substituting this vev in the potential, we can write the effective tree level potential for the φ 1,2 fields at low energies to be: where If we keep only the neutral components of the Higgs doublets, then the form of the potential is where Λ ≫ v wk . It is clear that although CP is spontaneously broken at a high scale Λ, at low energies, one has CP explicitly softly broken [9] by the bilinear terms in λ 12 . Note that both the fields φ 1,2 have Yukawa couplings and we can make a redefinition of the phase of one of the doublet fields (say φ 2 ) i.e. φ 2 → e −iβ φ 2 so that all the bilinear and O(Λ 2 ) terms in the potential become phase independent but the Yukawa couplings become complex. Thus the effective theory at low energies looks naively like hard CP violation, even though it is spontaneous CP violation at a very high scale. The Yukawa coupling Lagrangian looks like This still does not imply a viable complex CKM matrix; to achieve that, we must show that the vev of φ 2 where the phase resides, does not become very tiny when we demand the suppression of FCNC. In order to show this, let us write down the extremization of the potential as in section 2: For simplicity, we keep only the λ 1111 , λ 2222 and λ 1122 terms in the potential but our results follow in general: From these two equations, we find that both v 1 and v 2 are in general of the same order regardless of what the neutral Higgs masses are. This gives the CKM CP violation. As far as the masses of the neutral Higgs fields go, we can fine tune one set of parameters to keep one Higgs field light i.e. ≪ Λ and another will remain heavy thus suppressing the FCNC effects.
Of course we do not need to make the rephasing φ 2 → e −iβ φ 2 and eliminate the phase from the bilinear terms. If we do not do rephasing, the extremum equation of the Higgs potential would look like: Since Λ 2 ≫ v 2 , it is clear that to an excellent approximation one has: The phase δ would then appear in the quark mass matrices which will be nontrivially complex, thus leading to a complex CKM matrix. In Appendix B, we discuss how the fine tuning needed to keep the standard model Higgs at the electroweak scale does not prevent the components of the extra Higgs become superheavy in order to suppress the FCNC effects.

IV. SCPV WITHOUT FCNC PROBLEM IN FERMIONIC EXTENSIONS OF STANDARD MODEL
In this section, we briefly review the model in [10] in which an extension of the standard Higgs field σ as in sec. 3. The potential for the σ field is the same as in equation (11). As a result, the σ field has a complex vev leading to high scale spontaneous CP violation (since The CP violation is transmitted to the weak scale via its couplings given below: where g a , g ′ a , f, f ′ are real due CP conservation. But after symmetry breaking, the mass matrix contains terms mixing the heavy D quarks with the light d quarks [10]. This can be seen by writing down the full down quark mass matrix (in the notationψ L M dD ψ R ): where g and g ′ denote the row vectors (g 1 , g 2 , g 3 ) and (g ′ 1 , g ′ 2 , g ′ 3 ). Diagonalizing M dD M † dD , we can get the generalized 4 × 4 CKM matrix which indeed has a complex phase in the 3 × 3 sector involving the standard model quarks even in the limit of heavy D quark masses. This is an example of a breakdown of the decoupling theorem [10]. Clearly since there is only one neutral Higgs boson coupling to the effective down quark mass matrix, there is no  [13] and Higgs belonging to bidoublet field Φ(2, 2, 0) and a pair of fields of either (χ L (2, 1, −1)⊕χ R (1, 2, −1)) type (called χ-type below) or (∆ L (3, 1, +2) ⊕ ∆ R (1, 3, +2)) type (called ∆-type below). The left-right symmetric models are ideally suited to embed the first class of high scale SCPV models since the bidoublet Higgs field already contains the necessary two standard model doublet Higgs fields in it. All we have to do is to embed the high scale singlet field into a left-right Higgs field. We present two different ways to do this embedding in the two subsections below: A. Left-Right SCPV: Model I The first way to implement high scale SCPV is by choosing two pairs of χ type or ∆-type fields. Two pairs are needed since with a single pair, constraint that W R scale must be much higher than W L scale suppresses the SCPV phase by a factor M W L /M W R [14]. The two ∆ type model has been discussed in [11] where at the high scale, the ∆ R 's have vevs as follows: < ∆ 0 1,R > = v 1,R and < ∆ 2,R > = v 2,R e iδ . The coupling of the form T r(Φ † τ 2 Φ * τ 2 )T r(∆ † 1,R ∆ 2,R ) then induces the term λ 12 e iδ Λ 2 φ * 1 φ 2 at low energies and the rest of the discussion is as in section 3 above.
Let us now turn our attention to embedding of the model of Ref. [10] into the leftright model. We consider the left-right model without the bidoublet but with the (χ L (2, 1, −1) ⊕ χ R (1, 2, −1)) pair and three pairs of SU(2) L × SU(2) R singlet vector-like quarks (P L,R (1, 1, 4/3) and N L,R (1, 1, −2/3)). Such models were extensively studied in the early 90's but not from the point of view of spontaneous CP violation [12]. We take a complex singlet Higgs field σ as before and assume the theory to be CP conserving prior to symmetry breaking so that all couplings in the theory are real. Again, we assume the potential for the σ field to be as in Eq.11 so that its minimum corresponds to a complex vev for < σ >= Λe iδ as before. The vevs for the fields χ L,R are real.
To study the implications of the theory for low energy quark mixings, let us write down the quark Yukawa couplings: After spontaneous symmetry breaking we have < σ >= Λe iδ , < χ 0 L,R >= v L,R with v R ∼ Λ ≫ v L . This leads to the mass matrix of the form: Left-right symmetry requires that M P,N = M † P,N whereas the matrices h u,d are real. After diagonalization, the effective up and down quark mass matrices become: These matrices are hermitean and therefore lead to equal left and right handed CKM matrices as in the usual left-right models with bi-doublets and lead to complex CKM matrices.
In fact one can write the rotation matrices for both the up and down sector as follows in a basis where the couplings h u,d are diagonal: Clearly since U P,N is a unitary matrix with complex phases, V u,d will lead to complex CKM As far as the FCNC effects are concerned, they arise only in order m u,d /M P,N and therefore suppressed when Λ → large values. Note however that the quark mixing effects arise in zeroth order of this parameter.

B. Left-Right SCPV Model II:Connecting the CP violation and seesaw scales
In this subsection, we present a more economical left-right embedding of the high scale spontaneous CP violation with suppressed FCNC. The model consists of the usual left-right assignment of the fermions [13] and Higgs system consists of a single bidoublet φ(2, 2, 0) and the χ L (2, 1, −1) ⊕ χ R (1, 2, −1). Here spontaneous CP violation is implemented via the vev of a CP and P odd real singlet scalar field η [15]. The CP invariant Higgs potential for the theory can be written as: where with (a,b) going over (1,2) with φ 1 = φ and φ 2 = τ 2 φ * τ 2 .
We have assumed that under CP transformation η → −η and χ L → χ † R and φ → φ † . Invariance under this transformation requires that all parameters in the potential be real (except for one imaginary coupling shown explicitly in the above equation).
Note now that if the term in the potential connecting η and χ fields was absent, we would have < η >= 0 since M 2 η > 0. However as soon as SU(2) R symmetry is broken by < χ 0 R > = 0, the M ′ η term in the potential introduces a tadpole term for η thereby generating Since η is CP odd, this breaks CP spontaneously. The way it manifests is that the iµ < η > T r(φ † 1 φ 2 ) term now combines with the µ 2 12 T rφ † 1 φ 2 to generate at low energies an effective soft CP breaking term as in Eq. This can also be seen in an alternative manner by minimizing the potential, noting that there is a range of values of the parameters in the potential for which we have < χ 0 The vevs of χ R and η fields are much larger than the weak scale.
An interesting point worth stressing is that in this model, the scale of CP violation and the seesaw [16] scale for neutrino masses are connected. To see this, note that the right handed neutrino masses come from the higher dimensional term (L RχR ) 2 /M P l leading to seesaw right handed neutrino masses given by M seesaw ≃  [17]. A simple solution to this problem would of course be to have CP spontaneously broken. However, in MSSM, CP cannot be spontaneously broken.
Furthermore, it has also been pointed out that [18] it is particularly hard to have spontaneous CP breaking by considering multi-Higgs generalizations of the MSSM. A possibility for achieving spontaneous CP breaking within SUSY involves the introduction of singlet chiral fields [19]. As far as the FCNC effects are concerned, in these models one may fine tune the µ terms to make of the extra Higgs doublets heavy thereby eliminating large FCNC effects. However, the early versions of these models are no longer viable since they had a real CKM matrix, in contradiction with recent experimental data. Therefore, the ideas described in this paper may be particularly useful if one wants to solve the SUSY CP problem by spontaneous CP violation in a viable scenario, where vacuum phases do lead to a complex CKM matrix, while at the same time suppressing FCNC effects. In fact, recently it has been suggested one such model which includes two singlet Higgs superfields and adds an extra vector like singlet fermion to MSSM [20] to break CP spontaneously and generate a complex CKM matrix. One can embed this scheme into the SUSY left-right model. Detailed analysis of SUSY models that exploit the ideas of this paper is under way and will be taken up in a forthcoming publication.

VII. CONCLUSION
We have emphasized the close connection between spontaneous CP violation and FCNC effects in theories where CP breaking vev is at the weak scale. We have also shown that in order to avoid FCNC effects while at the same time generating a complex CKM matrix through vecuum phases, one is naturally led to have spontaneous CP breaking at a high energy scale, well above the electroweak scale.We then describe two classes of models one without and one with extra heavy fermions where having a high vev break CP spontaneously leads to complex CKM matrix as given by experiment without simultaneously having large FCNC effects. We then show how these models can be embedded into the high scale left-right models where parity violation and neutrino mass are connected via the seesaw mechanism.
We find one particular model where spontaneous parity violation triggers the spontaneous CP violation thus connecting the three scales: seesaw scale for neutrino masses, scale of spontaneous parity and CP violation.
In conclusion, if our view on the origin of CP violation is correct, then small neutrino masses and CP violation at low energies would have in common the fact that they are both manifestations of physics occuring at very high energy scale.

Appendix A
In this appendix, we elaborate on the connection between SCPV and FCNC and complex CKM in a two Higgs doublet model. For this purpose, we write the Yukawa Lagrangian as: It can be readily seen [21], [5] that in the quark mass eigenstate basis, the scalar coupling can be written as: where U d L,R are the unitary matrices which diagonalize the down quark mass matrix M d .
Analogous expressions are there for N u . It is clear that N d,u are in general not diagonal and therefore R and I mediate FCNC.
The quark mass matrices are in the form where H real is a symmetric real matrix. It is clear that M d M † d (and similarly M u M † u ) is an arbitray complex matrix and therefore CKM is a complex matrix.
If one fine tunes such that Y d 1 ∝ Y 2 d , N d would be diagonal and FCNC would be eliminated. But in that case, Eq(34) implies that M d M † d becomes real. This illustrates the connection between FCNC and the possibility of generating a complex CKM by a vacuum phase.